Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

TL;DR

This paper formulates a time-fractional KdV equation using variational methods, derives it with fractional derivatives, and solves it with an iterative approach to explore solitary wave properties.

Contribution

It introduces a novel derivation of the time-fractional KdV equation using variational principles and fractional calculus, and solves it with an iterative method.

Findings

Fractional derivatives influence solitary wave formation.

The derived equation captures effects of nonlinearity and dispersion.

Solutions reveal deeper insights into wave behavior with fractional calculus.

Abstract

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.